FWF - Sensornetzwerk Seminar - Computational Geometry Seminar with Applications to Sensor Networks

Project: Research project

Description

Geometric objects such as points, lines and polygons are the key elements of a big variety of interesting research problems in computer science. With the rise of modern technologies, more and more of these tasks are solved by computers, as opposed to the classic pen-and-paper approach.
Over the last thirty years, researchers around the world have developed different techniques and algorithms that take advantage of the structure provided by geometry to sIn this thesis we consider triangles in the colored Euclidean plane.
We call a triangle monochromatic if all its vertices have the same color.
First, we study how many colors are needed so that for every triangle we can color the Euclidean plane in such a way, that there does not exist a monochromatic rotated copy of the triangle or a monochromatic translated copy of the triangle.
Furthermore, we show that for every triangle every coloring of the Euclidean plane in finitely many colors contains a monochromatic triangle, which is similar to the given triangle.
Then we study the problem, for which triangles there exists a 6-coloring, such that the triangle is nonmonochromatic in this 6-coloring.
We also show, that for every triangle there exists a 2-coloring of the rational plane, such that the triangle is nonmonochromatic.
Finally we give a 5-coloring of a strip with height 1, such that there do not exist two points with distance 1, which have the same color.

olve these problems.
This area of research, in between mathematics and computer science, is known as discrete and computational geometry. In this joint seminar we plan to use tools from discrete aIn this thesis we consider triangles in the colored Euclidean plane.
We call a triangle monochromatic if all its vertices have the same color.
First, we study how many colors are needed so that for every triangle we can color the Euclidean plane in such a way, that there does not exist a monochromatic rotated copy of the triangle or a monochromatic translated copy of the triangle.
Furthermore, we show that for every triangle every coloring of the Euclidean plane in finitely many colors contains a monochromatic triangle, which is similar to the given triangle.
Then we study the problem, for which triangles there exists a 6-coloring, such that the triangle is nonmonochromatic in this 6-coloring.
We also show, that for every triangle there exists a 2-coloring of the rational plane, such that the triangle is nonmonochromatic.
Finally we give a 5-coloring of a strip with height 1, such that there do not exist two points with distance 1, which have the same color.

Computational geometry (such as order-type-like properties, see below) and apply them to problems that
come motivated from the field of sensor networks.
Status Finished 1/07/18 → 21/12/18